Given:
Focus of the parabola = (5,6)
Directrix of the parabola is y=2.
To find:
The equation of the parabola.
Solution:
The equation of the parabola is:
[tex](x-h)^2=4p(y-k)[/tex] ...(i)
Where, (h,k) is vertex, (h,k+p) is focus and [tex]y=k-p[/tex] is the directrix.
It is given that the focus of the parabola is at (5,6).
[tex](h,k+p)=(5,6)[/tex]
On comparing both sides, we get
[tex]h=5[/tex]
[tex]k+p=6[/tex] ...(ii)
Directrix of the parabola is y=2. So,
[tex]k-p=2[/tex] ...(iii)
Adding (ii) and (iii), we get
[tex]2k=8[/tex]
[tex]k=\dfrac{8}{2}[/tex]
[tex]k=4[/tex]
Putting [tex]k=4[/tex] in (ii), we get
[tex]4+p=6[/tex]
[tex]p=6-4[/tex]
[tex]p=2[/tex]
Putting [tex]h=5,k=4, p=2[/tex] in (i), we get
[tex](x-5)^2=4(2)(y-4)[/tex]
[tex](x-5)^2=8(y-4)[/tex]
Therefore, the equation of the parabola is [tex](x-5)^2=8(y-4)[/tex].
